respectively. They have poles and zeros that are mirror images with respect to the imaginary line \(\jmath \mathbb {R}\) , and have scaled coefficient matrices \(\jmath ^i R_i\)  that are Hermitian and skew-Hermitian, respectively. The nomenclature introduced above is used in the linear systems and control theory literature (see, for instance, [1]}, [2]}, [3]} and the references therein). However, in standard references on structured polynomial matrices [4]}, [5]} the para-Hermitian and para-skew-Hermitian structures are called alternating structures, because the matrix coefficients satisfy, respectively, \(R_i^* = (-1)^i R_i\)  and \(R_i^* = (-1)^{i+1} R_i\)  and, thus, alternate between being Hermitian or skew-Hermitian matrices. Specifically, para-Hermitian polynomial matrices are called \(*\) -even and para-skew-Hermitian polynomial matrices are called \(*\) -odd in [4]}, [5]}.
